C-Integrability Test for Discrete Equations via Multiple Scale Expansions

In this paper, we are extending the well-known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example, we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete...

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Дата:	2010
Автори:	Scimiterna, C., Levi, D.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2010
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/146506
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Цитувати:	C-Integrability Test for Discrete Equations via Multiple Scale Expansions / C. Scimiterna, D. Levi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 27 назв. — англ.

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spelling	irk-123456789-1465062019-02-10T01:24:35Z C-Integrability Test for Discrete Equations via Multiple Scale Expansions Scimiterna, C. Levi, D. In this paper, we are extending the well-known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example, we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential-difference equation. In this case, the equation satisfies the A₁, A₂ and A₃ linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable. 2010 Article C-Integrability Test for Discrete Equations via Multiple Scale Expansions / C. Scimiterna, D. Levi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34K99; 34E13; 37K10; 37J30 DOI:10.3842/SIGMA.2010.070 http://dspace.nbuv.gov.ua/handle/123456789/146506 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper, we are extending the well-known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example, we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential-difference equation. In this case, the equation satisfies the A₁, A₂ and A₃ linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable.
format	Article
author	Scimiterna, C. Levi, D.
spellingShingle	Scimiterna, C. Levi, D. C-Integrability Test for Discrete Equations via Multiple Scale Expansions Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Scimiterna, C. Levi, D.
author_sort	Scimiterna, C.
title	C-Integrability Test for Discrete Equations via Multiple Scale Expansions
title_short	C-Integrability Test for Discrete Equations via Multiple Scale Expansions
title_full	C-Integrability Test for Discrete Equations via Multiple Scale Expansions
title_fullStr	C-Integrability Test for Discrete Equations via Multiple Scale Expansions
title_full_unstemmed	C-Integrability Test for Discrete Equations via Multiple Scale Expansions
title_sort	c-integrability test for discrete equations via multiple scale expansions
publisher	Інститут математики НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/146506
citation_txt	C-Integrability Test for Discrete Equations via Multiple Scale Expansions / C. Scimiterna, D. Levi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 27 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT scimiternac cintegrabilitytestfordiscreteequationsviamultiplescaleexpansions AT levid cintegrabilitytestfordiscreteequationsviamultiplescaleexpansions
first_indexed	2025-07-11T00:08:30Z
last_indexed	2025-07-11T00:08:30Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 070, 17 pages C-Integrability Test for Discrete Equations via Multiple Scale Expansions Christian SCIMITERNA and Decio LEVI Dipartimento di Ingegneria Elettronica, Università degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy E-mail: scimiterna@fis.uniroma3.it, levi@roma3.infn.it Received May 29, 2010, in final form August 20, 2010; Published online August 31, 2010 doi:10.3842/SIGMA.2010.070 Abstract. In this paper we are extending the well known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf–Cole transformation reduces to a linear differential difference equation. In this case the equation satisfies the A1, A2 and A3 linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non- secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable. Key words: linearizable discrete equations; linearizability theorem; multiple scale expansion; obstructions to linearizability; discrete Burgers 2010 Mathematics Subject Classification: 34K99; 34E13; 37K10; 37J30 1 Introduction Calogero in 1991 [3] introduced the notion of S and C integrable equations to denote those nonlinear partial differential equations which are solvable through an inverse Spectral transform or linearizable through a Change of variables. Using the multiple scale reductive technique he was able to show that the Nonlinear Schrödinger Equation (NLSE) i∂tf + ∂xxf = ρ2\|f \|2f, f = f(x, t), (1.1) appears as a universal equation governing the evolution of slowly varying packets of quasi- monochromatic waves in weakly nonlinear media featuring dispersion. Calogero and Eckhaus then showed that a necessary condition for the S-integrability of a dispersive nonlinear partial differential equation is that its multiple scale expansion around a slowly varying packet of quasi-monochromatic wave should provide at the lowest order in the perturbation parameter an integrable NLSE. Then, by a paradox, they showed that a C-integrable equation must reduce to a linear equation or another C-integrable equation as the Eckhaus equation [4, 22]. In the case of discrete equations it has been shown [26, 1, 16, 18, 17, 19, 10, 11, 12] that a similar situation is also true. One presents the equivalent of the Calogero–Eckhaus theorem stating that a necessary condition for a nonlinear dispersive partial difference equation to be S-integrable is that the lowest order multiple scale expansion on C(∞) functions give rise to integrable NLSE. The nonlinear dispersive partial difference equation will be C-integrable if its multiple scale expansion on C(∞) functions will give rise at the lowest order to a linear or a C-integrable differential equation. By going to a higher order in the expansion, the multiple scale techniques give more stringent conditions which have been used to find new S-integrable Partial Differential Equations (PDE’s) mailto:scimiterna@fis.uniroma3.it mailto:levi@roma3.infn.it http://dx.doi.org/10.3842/SIGMA.2010.070 2 C. Scimiterna and D. Levi and to prove the integrability of new nonlinear equations [8, 9, 15]. Probably the most important example of such nonlinear PDE is the Degasperis–Procesi equation [7]. Up to our knowledge higher order expansions for nonlinear linearizable equations have not been considered in details. The purpose of this paper is to show that the integrability theorem, stated in [11], can be extended to the case of linearizable difference equations, providing a way to discriminate between S-integrable, C-integrable and non–integrable lattice equations. The continuous (and thus discrete) higher order C-integrability conditions are, up to our knowledge, presented here for the first time. We apply here the resulting linearizability conditions to a differential-difference dispersive nonlinear equation of the discrete Burgers hierarchy [21] and its difference-difference analogue. In Section 2 we present the differential-difference linearizable nonlinear dispersive Burgers and its partial difference analogue and discuss the tools necessary to carry out the multiple scale C-integrability test. In Section 3 we apply them to the two equations previously introduced, leaving to the Appendix all details of the calculation, and present in Section 4 some conclusive remarks. 2 Multiple scale perturbation reduction of Burgers equations The Burgers equation, the simplest nonlinear equation for the study of gas dynamics with heat conduction and viscous effect, was introduced by Burgers in 1948 [2, 27]. Explicit solutions of the Cauchy problem on the infinite line for the Burgers equation may be obtained by the Hopf– Cole transform, introduced independently by Hopf and Cole in 1950 [5, 14]. This transformation linearize the equation and the solution of the linearized equation provide solutions of the Burgers equation. Bruschi, Levi and Ragnisco [21] extended the Hopf–Cole transformation to construct hier- archies of linearizable nonlinear matrix PDE’s, nonlinear differential-difference equations and difference-difference equations. The simplest differential–difference nonlinear dispersive equation of the Burgers hierarchy is given by ∂tun (t) = 1 2h { [1 + hun (t)] [un+1 (t)− un (t)]− un−1 (t)− un (t) 1 + hun−1 (t) } , (2.1) where the function un(t) and the lattice parameter h are all supposed to be real. Equation (2.1) has a nonlinear dispersion relation ω = − sin(κh) h . When h → 0 and n → ∞ in such a way that x = nh is finite, the Burgers equation (2.1) reduces to the one dimensional wave equation ∂tu− ∂xu = O(h3). Through the discrete Cole–Hopf transformation un (t) = φn+1 (t)− φn (t) hφn (t) (2.2) equation (2.1) linearizes to the discrete linear wave equation ∂tφn (t) = φn+1 (t)− φn−1 (t) 2h . (2.3) The transformation (2.2) can be inverted and gives φn = φn0 j=n−1∏ j=n0 (1 + huj) , n ≥ n0 + 1, (2.4a) C-Integrability Test for Discrete Equations via Multiple Scale Expansions 3 φn = φn0 j=n0−1∏ j=n (1 + huj) , n ≤ n0 − 1, (2.4b) where φn0 = φn0(t) is the function φn calculated at a given initial point n = n0. When un(t) satisfies equation (2.1), the function φn(t) will satisfy the discrete wave equation (2.3) if φn0 satisfies the ordinary differential equation φ̇n − 1 2h [ 1 + hun − 1 (1 + hun−1) ] φn ∣∣∣ n=n0 = 0, whose solution is given by φn0(t) = φn0(t0) exp { 1 2h ∫ t t0 [ 1 + hun − 1 (1 + hun−1) ] ∣∣∣ n=n0 dt′ } , and t0 is an initial time. If lim n→−∞ un(t) = u−∞ is finite equations (2.4) reduce to φn = α (t) (1 + hu−∞)n γ=n−1∏ γ=−∞ ( 1 + huγ 1 + hu−∞ ) , where α (t) is a t-dependent function, α(t) = α0 exp { 1 2 ( 1 + 1 1 + hu−∞ ) u−∞t } . A C-integrable discretization of equation (2.1) is given by the partial difference equation un,m+1 − un,m σ = 1 2h [ (1 + hun,m)(un+1,m − un,m+1)− un−1,m − un,m+1 1 + hun−1,m ] , (2.5) where σ is the constant lattice parameter in the time variable. Equation (2.5) is dissipative as it has a complex dispersion relation ω = i σ ln [ 1 + iσh sin (κh) ] . As we are not able to con- struct a dispersive counterpart of equation (2.1) we consider a straightforward discretization of equation (2.1) un,m+1 − un,m−1 2σ = 1 2h [ (1 + hun,m) (un+1,m − un,m)− (un−1,m − un,m) 1 + hun−1,m ] , (2.6) whose nonlinear dispersion relation is sin (ωσ) = −sin (κh)σ h . In the remaining part of this section we will present the tools necessary to carry out the multiple scale expansion of equations (2.1), (2.6) and construct the conditions which they must satisfy to be C-integrable equation of j order, with j = 1, 2, 3, i.e. such that asymptotically they reduce to a linear equation up to terms respectively of the third, fourth and fifth order in the perturbation paramether. These conditions up to our knowledge have been presented for the first time by Dr. Scimiterna in his PhD Thesis [25] and are published here for the first time. 2.1 Expansion of real dispersive partial difference equations For completeness we briefly illustrate here all the ingredients of the reductive perturbative technique necessary to treat difference equations, as presented in [10, 11, 25]. 4 C. Scimiterna and D. Levi 2.1.1 From shifts to derivatives Let us consider a function un : Z → R depending on an index n ∈ Z and let us suppose that: • The dependence of un on n is realized through the slow variable n1 .= εn ∈ R, ε ∈ R, 0 < ε� 1, that is to say un .= u(n1). • The function u (n1) ∈ C(∞) (D), where D ∈ R is a region containing the point n1. Under these hypotheses one can write the action of the shift operator Tn such that Tnun .= un+1 = u(n1 + ε) as the following (formal) series Tnu(n1) = u(n1) + εu,n1(n1) + ε2 2 u,2 n1(n1) + · · ·+ εk k! u,k n1(n1) + · · · = +∞∑ k=0 εk k! u,k n1(n1), (2.7) where u,k n1(n1) .= dku(n1)/dnk 1 .= dk n1 u(n1), being dn1 the total derivative operator. The last expression suggests the following formal expansion for the differential operator Tn: Tn = +∞∑ k=0 εk k! dk n1 .= eεdn1 valid only when the series in equation (2.7) is converging. So we must require that the radius of convergence of the series starting at n1 is wide enough to include as an inner point at least the point n1 + ε. Let us introduce more complicated dependencies of un on n. For example one can assume a simultaneous dependence on the fast variable n and on the slow variable n1, i.e. un .= u(n, n1). The action of the total shift operator Tn will now be given by Tnun .= un+1 = u(n + 1, n1 + ε) so that we can write Tn .= T (1) n T (ε) n1 , where the partial shift operators T (1) n and T (ε) n1 are defined respectively by T (1) n u(n, n1) = u(n+ 1, n1) = ∞∑ k=0 1 k! ∂k nu(n, n1) = e∂nu(n, n1), and T (ε) n1 u(n, n1) = u(n, n1 + ε) = ∞∑ k=0 εk k! ∂k n1 u(n, n1) = eε∂n1u(n, n1). (2.8) The dependence of un on n can be easily extended to the case of one fast variable n and K slow variables nj .= εjn, εj ∈ R, 1 ≤ j ≤ K each of them being defined by its own parameter εj . The action of the total shift operator Tn will now be given in terms of the partial shifts T (1) n , T (ε) nj , as Tn .= T (1) n K∏ j=1 T (εj) nj . Let us now consider a nonlinear partial difference equation F [ {un+k,m+j} j=(−K(−),K(+)) k=(−N (−),N (+)) ] = 0, ( N (±),K(±) ) ≥ 0, (2.9) for a function un,m : Z2 → R which now depends on two indexes n and m ∈ Z which we will term respectively as discrete space and time indices. Equation (2.9) contains m and n-shifts, respectively in the intervals (m−K(−),m+K(+)) and (n−N (−), n+N (+)). Under some obvious C-Integrability Test for Discrete Equations via Multiple Scale Expansions 5 Table 1. The operators A(j) n , B(j) m and C(j) n,m appearing in equations (2.10). j = 0 j = 1 j = 2 j = 3 j = 4 A(j) n 1 N1∂n1 N2 1 2 ∂2 n1 N3 1 6 ∂3 n1 N4 1 24 ∂4 n1 B(j) m 1 M1∂m1 M2 1 2 ∂2 m1 + M2∂m2 M3 1 6 ∂3 m1+ M4 1 24 ∂4 m1 + M2 1 M2 2 ∂2 m1∂m2+ +M1M2∂m1∂m2+ + M2 2 2 ∂2 m2 + M1M3∂m1∂m3 + M4∂m4 +M3∂m3 C(j) n,m 1 A(1) n + B(1) m A(2) n + B(2) m + A(3) n + B(3) m + A(4) n + B(4) m + +N1M1∂n1∂m1 +N1M2∂n1∂m2+ + M3 1 N1 6 ∂3 m1∂n1 + N3 1 M1 6 ∂3 n1∂m1+ + M1N2 1 2 ∂2 n1∂m1+ +N1M1M2∂n1∂m1∂m2 + N2 1 M2 2 ∂2 n1∂m2+ + N1M2 1 2 ∂n1∂2 m1 + N2 1 M2 1 4 ∂2 n1∂2 m1 + N1M3∂n1∂m3 hypothesis on the C(∞) property of the function un,m and on the radius of convergence of its Taylor expansion for all shifts in the indices n and m involved in the difference equation (2.9), we can write a series representation of un+k,m+j around un,m. We choose the slow variables as nk .= εnk n, mj .= εmjm with εnk .= Nkε k, 1 ≤ k ≤ Kn, εmj .= Mjε j , 1 ≤ j ≤ Km, where the various constants Nk, Mj and ε are all real numbers. In this presentation we assume Kn = 1 and Km = K (eventually K = +∞) so that Tn = T (1) n T (εn1 ) n1 = T (1) n +∞∑ j=0 εjA(j) n , (2.10a) Tm = T (1) m K∏ j=1 T (εmj ) mj = T (1) m +∞∑ j=0 εjB(j) m , (2.10b) TnTm = T (1) n T (1) m T (εn1 ) n1 K∏ j=1 T (εmj ) mj = T (1) n T (1) m +∞∑ j=0 εjC(j) n,m, (2.10c) where the operators A(j) n , B(j) m , and C(j) n,m are given in Table 1. Inserting the explicit expres- sions (2.10) of the shift operators into equation (2.9), this turns into a PDE of infinite order. So we will assume for the function un,m = u(n,m, n1, {mj}K j=1 , ε) a double expansion in harmonics and in the perturbative parameter ε un,m = +∞∑ γ=1 γ∑ θ=−γ εγu(θ) γ (n1,mj , j ≥ 1) eiθ(κhn−ω(κ)σm), (2.11) with u(−θ) γ (n1,mj , j ≥ 1) = ū (θ) γ (n1,mj , j ≥ 1), where by a bar we denote the complex conjugate, in order to ensure the reality of un,m. The index γ is chosen ≥ 1 so that the nonlinear terms of equation (2.9) enter as a perturbation in the multiple scale expansion. For simplicity we will set N1 = Mj = 1, j ≥ 1. Moreover we will assume that the functions u(θ) γ satisfy the asymptotic conditions lim n1→±∞ u (θ) γ = 0, ∀ γ and θ to provide a meaningful expansion. 2.1.2 From derivatives to shifts The multiple scale approach discussed above reduces a given partial difference equation into a partial differential equation for the amplitudes u(θ) γ contained in the definition (2.11). 6 C. Scimiterna and D. Levi We can rewrite the so obtained partial differential equation as a partial difference equa- tion inverting the expansion of the partial shift operator in term of partial derivatives (2.8). From (2.8) we have ∂n1 = 1 ε ln T (ε) n1 = 1 ε ln ( 1 + ε∆(+) n1 ) .= +∞∑ k=1 (−ε)k−1 k [ ∆(+) n1 ]k , (2.12) where ∆(+) n1 .= (T (ε) n1 − 1)/ε is the first forward difference operator with respect to the slow- variable n1. This is just one of the possible inversion formulae for the operator T (ε) n1 . For example an expression similar to equation (2.12) can be written for the first backward difference operator ∆(−) n1 .= ( 1− [ T (ε) n1 ]−1) /ε. For the first symmetric difference operator ∆(s) n1 .= ( T (ε) n1 − [ T (ε) n1 ]−1) /2ε we get ∂n1 = sinh−1 ε∆(s) n1 .= +∞∑ k=1 Pk−1(0)εk k [ ∆(s) n1 ]k , where Pk(0) is the k-th Legendre polynomial evaluated at x = 0. Only when we impose that the function un is a slow-varying function of order ` in the variable n1, i.e. that ∆`+1 n1 un = 0, we can see that the ∂n1 operator, which is given by formal series containing in general infinite powers of the ∆n1 , reduces to polynomial of order at most `. In [17], choosing ` = 2 for the indexes n1 and m1 and ` = 1 for m2, it was shown that the integrable lattice potential KdV equation [20] reduces to a completely discrete and local nonlinear Schrödinger equation which has been proved to be not integrable by singularity confinement and algebraic entropy [13, 23]. Consequently, if one passes from derivatives to shifts, one ends up in general with a nonlocal partial difference equation in the slow variables nκ and mδ. 2.2 The orders beyond the Schrödinger equation and the C-integrability conditions The multiple scale expansion of an equation of the Burgers hierarchy on functions of infinite order will thus give rise to PDE’s. So a multiple scale integrability test will require that a dis- persive equation like equation (2.1) is C-integrable if its multiple scale expansion will go into the hierarchy of the Schrödinger equation i∂tψ + ∂2 xψ = 0. To be able to verify the C-integrability we need to consider in principle all the orders beyond the Schrödinger equation. This in general will not be possible but already a few orders beyond the Schrödinger equation might be sufficient to verify if the equation is linearizable or not. In the case of S-integrable nonlinear PDE’s the first attempt to go beyond the NLSE order has been presented by Degasperis, Manakov and Santini in [8]. These authors, starting from an S-integrable model, through a combination of an asymptotic functional analysis and spectral methods, succeeded in removing all the secular terms from the reduced equations order by order. Their results could be summarized as follows: 1. The number of slow-time variables required for the amplitudes u(θ) j appearing in (2.11) coincides with the number of nonvanishing coefficients of the Taylor expansion of the dispersion relation, ωj (κ) = 1 j! djω(k) dkj . 2. The amplitude u(1) 1 evolves at the slow-times ms, s ≥ 2 according to the s-th equation of the NLS hierarchy. C-Integrability Test for Discrete Equations via Multiple Scale Expansions 7 3. The amplitudes of the higher perturbations of the first harmonic u(1) j , j ≥ 2 evolve, taking into account some asymptotic boundary conditions, at the slow-times ms, s ≥ 2 according to certain linear, nonhomogeneous equations. Then they concluded that the cancellation at each stage of the perturbation process of all the secular terms is a sufficient condition to uniquely fix the evolution equations followed by every u(1) j , j ≥ 1 for each slow-time ms. Point 2 implies that a hierarchy of integrable equations provide for a function u always compatible evolutions, i.e. the equations in its hierarchy are generalized symmetries of each other. In this way this procedure provides necessary and sufficient conditions to get secularity-free reduced equations [8]. We apply the present procedure to the case of C-integrable partial difference equations. Following Degasperis and Procesi [9] we state the following theorem: Theorem 1. If equation (2.9) is C-integrable then, after a multiple scale expansion, the func- tions u(1) j , j ≥ 1 satisfy the equations ∂msu (1) 1 − (−i)s−1Bs∂ s n1 u (1) 1 .= Msu (1) 1 = 0, (2.13a) Msu (1) j = fs(j), (2.13b) ∀ j, s ≥ 2, where Bs∂ s n1 u (j) 1 is the s-th f low in the linear Schrödinger hierarchy and Bs are real constants. All the other u(κ) j , κ ≥ 2 are expressed as differential monomials of u(1) r , r ≤ j − 1. In equation (2.13b) fs(j) is a nonhomogeneous nonlinear forcing written in term of differential monomials of u(1) r , r ≤ j. From Theorem 1 it follows that a nonlinear partial difference equation is said to be C-integrable if its asymptotic multiple scale expansion is given by a uniform asymptotic series whose leading harmonic u(1) possesses an infinity of generalized symmetries evolving at different times and given by commuting linear equations. Equations (2.13) are a necessary condition for C-integrability. It is worthwhile to stress here the non completely obvious fact that, in contrast to the first order wave equation, ∂tu = ∂xu, all the symmetries of the Schrödinger equation commuting with it are given by the equations (2.13a) and only by them. This implies that all the equations appearing in the multiple scale expansion for a C-integrable equation are uniquely defined. It is obvious that the operators Ms defined in equation (2.13a) commute among themselves. However the compatibility of equations (2.13b) is not always guaranteed but is subject to some compatibility conditions among their r.h.s. terms fs(j). Once we fix the index j ≥ 2 in the set of equations (2.13b), this commutativity condition implies the compatibility conditions Msfs′ (j) = Ms′fs (j) , ∀ s, s′ ≥ 2, (2.14) where, as fs (j) and fs′ (j) are functions of the different perturbations u(1) r of the fundamental harmonic up to degree j−1, the time derivatives ∂ms , ∂ms′ appearing respectively in Ms and Ms′ have to be eliminated using the evolution equations (2.13) up to the index j − 1. These last commutativity conditions turn out to be a linearizability test. Following [8] we conjecture that the relations (2.13) are a sufficient condition for the C- integrability or that the C-integrability is a necessary condition to have a multiple scale expan- sion where equations (2.13) are satisfied. To characterize the functions fs(j) we introduce the following definitions: Definition 1. A differential monomial M [ u (1) j ] , j ≥ 1 in the functions u(1) j , its complex conju- gate and its n1-derivatives is a monomial of “gauge” 1 if it possesses the transformation property M [ ũ (1) j ] = eiθM [ u (1) j ] , when ũ (1) j .= eiθu (1) j . 8 C. Scimiterna and D. Levi Definition 2. A finite dimensional vector space Pν , ν ≥ 2 is the set of all differential polyno- mials of gauge 1 in the functions u(1) j , j ≥ 1, their complex conjugates and their n1-derivatives such that their total order in ε is ν, i.e. order ( ∂µ n1 u (1) j ) = order ( ∂µ n1 ū (1) j ) = µ+ j = ν, µ ≥ 0. Definition 3. Pν(µ), µ ≥ 1 and ν ≥ 2 is the subspace of Pν whose elements are differential polynomials of gauge 1 in the functions u(1) j , their complex conjugates and their n1-derivatives such that their total order is ν and 1 ≤ j ≤ µ. From Definition 3 it follows that Pν = Pν(ν−2). Moreover in general fs(j) ∈ Pj+s(j−1) where j, s ≥ 2. The basis monomials of the spaces Pν(µ) in which we can express the func- tions fs(j) can be found, for example, in [25]. Proposition 1. If for each fixed j ≥ 2 the equation (2.14) with s = 2 and s′ = 3, namely M2f3 (j) = M3f2 (j) , (2.15) is satisfied, then there exist unique differential polynomials fs(j) ∀ s ≥ 4 such that the flows Msu (1) j = fs (j) commute for any s ≥ 2 [24, 6]. Hence among the relations (2.14) only those with s = 2 and s′ = 3 have to be tested. Proposition 2. The homogeneous equation Msu = 0 has no solution u in the vector space Pµ, i.e. Ker (Ms) ∩ Pµ = ∅. Consequently the multiple scale expansion (2.13) is secularity-free. This does not mean that, in solving equation (2.13b), we have to set to zero all the contributions to the solution coming from the homogeneous equation but only that part of it which is present in the forcing terms. Finally: Definition 4. If the relations (2.14) are satisfied up to the index j, j ≥ 2, we say that our equation is asymptotically C-integrable of degree j or Aj C-integrable. 2.2.1 Integrability conditions for the Schrödinger hierarchy Here we present the conditions for the asymptotic C-integrability of order k or Ak C-integrability conditions with k = 1, 2, 3. To simplify the notation, we will use for u(1) j the concise form u(j). The A1 C-integrability condition is given by the absence of the coefficient ρ2 of the nonlinear term in the NLSE (1.1). The A2 integrability conditions are obtained choosing j = 2 in the compatibility condi- tions (2.14) with s = 2 and s′ = 3 as in (2.15). In this case we have that f2(2) ∈ P4(1) and f3(2) ∈ P5(1) where P4(1) contains 2 different differential monomials and P5(1) contains 5 different differential monomials, so that f2(2) and f3(2) will be respectively identified by 2 and 5 complex constants f2(2) .= au,n1(1)\|u(1)\|2 + bū,n1(1)u(1)2, (2.16) f3(2) .= α\|u(1)\|4u(1) + β\|u,n1(1)\|2u(1) + γu,n1(1)2ū(1) + δū,2n1(1)u(1)2 + ε\|u(1)\|2u,2n1(1). In this way, eliminating from equation (2.15) the derivatives of u(1) with respect to the slow- times m2 and m3 using the evolutions (2.13a) with s = 2, 3 and equating term by term, we obtain that the A2 C-integrability conditions gives no constraints on the coefficients a and b C-Integrability Test for Discrete Equations via Multiple Scale Expansions 9 appearing in f2(2). The expression of the coefficients α, β, γ, δ, ε appearing in f3(2) in terms of a and b are α = 0, β = −3iB3b B2 , γ = −3iB3a 2B2 , δ = 0, ε = γ. The A3 C-integrability conditions are derived in a similar way setting j = 3 in equation (2.15). In this case we have that f2(3) ∈ P5(2) and f3(3) ∈ P6(2) where P5(2) contains 12 different differential monomials and P6(2) contains 26 different differential monomials, so that f2(3) and f3(3) will be respectively identified by 12 and 26 complex constants f2(3) .= τ1\|u(1)\|4u(1) + τ2\|u,n1(1)\|2u(1) + τ3\|u(1)\|2u,2n1(1) + τ4ū,2n1(1)u(1)2 + τ5u,n1(1)2ū(1) + τ6u,n1(2)\|u(1)\|2 + τ7ū,n1(2)u(1)2 + τ8u(2)2ū(1) (2.17) + τ9\|u(2)\|2u(1) + τ10u(2)u,n1(1)ū(1) + τ11u(2)ū,n1(1)u(1) + τ12ū(2)u,n1(1)u(1), f3(3) .= γ1\|u(1)\|4u,n1(1) + γ2\|u(1)\|2u(1)2ū,n1(1) + γ3\|u(1)\|2u,3n1(1) + γ4u(1)2ū,3n1(1) + γ5\|u,n1(1)\|2u,n1(1) + γ6ū,2n1(1)u,n1(1)u(1) + γ7u,2n1(1)ū,n1(1)u(1) + γ8u,2n1(1)u,n1(1)ū(1) + γ9\|u(1)\|4u(2) + γ10\|u(1)\|2u(1)2ū(2) + γ11ū,n1(1)u(2)2 + γ12u,n1(1)\|u(2)\|2 + γ13\|u,n1(1)\|2u(2) + γ14\|u(2)\|2u(2) + γ15u,n1(1)2ū(2) + γ16\|u(1)\|2u,2n1(2) + γ17u(1)2ū,2n1(2) + γ18u(2)ū,2n1(1)u(1) + γ19u(2)u,2n1(1)ū(1) + γ20ū(2)u,2n1(1)u(1) + γ21u(2)u,n1(2)ū(1) + γ22ū(2)u,n1(2)u(1) + γ23u,n1(2)u,n1(1)ū(1) + γ24u,n1(2)ū,n1(1)u(1) + γ25ū,n1(2)u,n1(1)u(1) + γ26ū,n1(2)u(2)u(1). Let us eliminate from equation (2.15) with j = 3 the derivatives of u(1) with respect to the slow- times m2 and m3 using the evolutions (2.13a) with s = 2, 3 and the same derivatives of u(2) using the evolutions (2.13b) with s = 2, 3. Equating the remaining terms term by term, the A3 C-integrability conditions turn out to be: τ1 = − i 4B2 [b (τ11 − 2τ6) + āτ7] , b̄τ7 = 1 2 (b− a) (τ11 + τ10 − τ6) + āτ7, aτ8 = bτ8 = 0, aτ9 = bτ9 = 0, āτ12 = a (τ10 − τ11) + bτ6 + āτ7,( b̄− ā ) τ12 = (b− a) τ10. (2.18) Sometimes a and b turn out to be both real. In this case the conditions given in equations (2.18) becomes: R1 = 1 4B2 [b (I11 − 2I6) + aI7] , I1 = − 1 4B2 [b (R11 − 2R6) + aR7] , (b− a) (R11 +R10 −R6 − 2R7) = 0, (b− a) (I11 + I10 − I6 − 2I7) = 0, (b− a)R8 = 0, (b− a) I8 = 0, (b− a)R9 = 0, (b− a) I9 = 0, a (R12 +R11 −R10 −R7) = bR6, a (I12 + I11 − I10 − I7) = bI6, (b− a) (R12 −R10) = 0, (b− a) (I12 − I10) = 0, (2.19) where τi = Ri + iIi for i = 1, . . . , 12. The expressions of the γj as functions of the τi are: γ1 = 3B3 4B2 2 ( aτ6 − 4iB2τ1 + b̄τ12 ) , γ2 = 3B3 4B2 2 (bτ6 + āτ7) , γ3 = −3iB3τ3 2B2 , γ4 = 0, γ5 = −3iB3τ2 2B2 , γ6 = −3iB3τ4 B2 , 10 C. Scimiterna and D. Levi γ7 = γ5, γ8 = γ3 − 3iB3τ5 B2 , γ9 = γ10 = γ11 = 0, γ12 = −3iB3τ9 2B2 , γ13 = −3iB3τ11 2B2 , γ14 = 0, γ15 = −3iB3τ12 2B2 , γ16 = −3iB3τ6 2B2 , γ17 = γ18 = 0, γ19 = −3iB3τ10 2B2 , γ20 = γ15, γ21 = −3iB3τ8 B2 , γ22 = γ12, γ23 = γ16 + γ19, γ24 = γ13, γ25 = −3iB3τ7 B2 , γ26 = 0. The conditions given in equations (2.18), (2.19) appear to be new. Their importance resides in the fact that a C-integrable equation must satisfy those conditions. 3 Linearizability of the equations of the Burgers hierarchy Taking into account the results of the previous section we can carry out the multiple scale expansion of the equations of the Burgers hierarchy. To do so we substitute the definition (2.11) into equations (2.1), (2.6) and write down the coefficients of the various harmonics θ and of the various orders j of ε. When we deal with the differential-difference equation (2.1), we have to make the substitutions σm → t, σmi → ti. This transformation implies that in this case the corresponding coefficients ρ2 and B2 will turn out to be σ-independent. In Appendix we present all relevant equations and here we just present their results. Proposition 3. The differential-difference equation (2.1) of the Burgers hierarchy satisfies the A1 (and consequently also the A2) and also the A3 C-integrability conditions. Proposition 4. The partial difference Burgers-like equation (2.6) reduces for j = 3 and α = 1 to a NLSE with a nonlinear complex coefficient ρ2 given by equation (A.9c). Thus the equation is neither S-integrable nor C-integrable. 4 Conclusions In the present paper we have presented all the steps necessary to apply the perturbative multiple scale expansion to dispersive nonlinear differential-difference or partial difference equations which may be linearizable. These passages involve the representation of the lattice variables in terms of an infinite set of derivative with respect to the lattice index and the analysis of the higher order of the perturbation which give rise to a set of compatible higher order linear PDE’s belonging to the hierarchy of the Schrödinger equation. The compatibility of these equations give rise to a linearizability test. We applied the so obtained test to the case of a differential- difference dispersive Burgers equation and its discretization. It turns out that the Burgers is linearizable (as it should be) but its discretization is neither S-integrable nor C-integrable. So, effectively this procedure is able to distinguish between linearizable and non-linearizable equations. A Appendix Let us now start performing a multiple scale analysis of the partial difference equation (2.6). We present here the equations we get at the various orders of ε and for the different harmonics θ. • Order ε and θ = 0: In this case the resulting equation is automatically satisfied. C-Integrability Test for Discrete Equations via Multiple Scale Expansions 11 • Order ε and θ = 1: If one requires that u(1) 1 6= 0, one obtains the dispersion relation sin (ωσ) = −sin (κh)σ h . (A.1) • Order ε2 and θ = 0: We obtain the evolution ∂m1u (0) 1 − σ h ∂n1u (0) 1 = 0, (A.2) which implies that u(0) 1 depends on the variable ρ .= hn1 + σm1. • Order ε2 and θ = 1: Taking into account the dispersion relation (A.1), we have ∂m1u (1) 1 − σ cos (ωσ) [ cos (κh) h ∂n1u (1) 1 − 2 sin2 ( κh 2 ) u (0) 1 u (1) 1 ] = 0, (A.3) which implies that u(1) 1 has the form u (1) 1 = g (ξ,mj , j ≥ 2) exp { δ ∫ ρ ρ0 u (0) 1 ( ρ′ ) dρ′ } , δ .= 2 sin2 (κh/2) [cos (κh)− cos (ωσ)] , (A.4) where ξ .= hn1 + cos(κh) cos(ωσ)σm1, g is an arbitrary function of its arguments going to zero as ξ → ±∞ and ρ0, by a proper redefinition of g, can always be chosen to be a zero of u(0) 1 as when ρ→ ±∞, u(0) 1 → 0 so that there will exists at least one zero. • Order ε2 and θ = 2: Taking into account the dispersion relation (A.1), we have u (2) 2 = ( e−iκh − 1 ) h 2 [cos (κh)− cos (ωσ)] [ u (1) 1 ]2 . (A.5) • Order ε3 and θ = 0: We have ∂m1u (0) 2 − σ h ∂n1u (0) 2 = −∂m2u (0) 1 − 2σ sin2 (κh/2) [ ∂n1 + 2hu(0) 1 ]∣∣u(1) 1 ∣∣2. (A.6) By equation (A.2), the term ∂m2u (0) 1 is a solution of the left hand side of equation (A.6), hence it is a secular term. As a consequence we have to require that ∂m1u (0) 2 − σ h ∂n1u (0) 2 = −2σ sin2 (κh/2) [ ∂n1 + 2hu(0) 1 ]∣∣u(1) 1 ∣∣2, ∂m2u (0) 1 = 0. Solving equation (A) taking into account equation (A.4), we obtain u (0) 2 = f (ρ,mj , j ≥ 2)− hδ cos (ωσ) [ \|u(1) 1 \|2 + 2 (1 + δ)u(0) 1 ∫ ξ ξ0 \|u(1) 1 \|2dξ′ ] , (A.7) where f is an arbitrary function of its arguments going to zero as ρ → ±∞ and ξ0 is an arbitrary value of the variable ξ. Let us restrict ourselves for simplicity to the case where there is no dependence at all on ρ. If one wants that the harmonic u(0) 1 depends on ξ and not on ρ, from equation (A.2) one has that ∂ξu (0) 1 = 0, so that u(0) 1 depends on the slow variables mj , j ≥ 2 only. Similarly we have that ∂ξf = 0. In this case, in order to satisfy the asymptotic conditions lim ξ→±∞ u (0) γ = 0, γ = 1, 2, one has to take u(0) 1 = f = 0 (unless we take the fully continuous limit h → 0, hn1 .= x1, σ → 0, σm1 .= t1 in which ρ → ξ). Equation (A.7) then becomes u (0) 2 = −hδ cos (ωσ) ∣∣u(1) 1 ∣∣2. (A.8) 12 C. Scimiterna and D. Levi • Order ε3 and θ = 1: Taking into account the dispersion relation (A.1) and the equa- tions u(0) 1 = 0, (A.3), (A.5), (A.8), we have ∂m1u (1) 2 − σ cos (κh) h cos (ωσ) ∂n1u (1) 2 = −∂m2u (1) 1 − iB2∂ 2 ξu (1) 1 − iρ2 ∣∣u(1) 1 ∣∣2u(1) 1 , (A.9a) B2 .= hσ ( h2 − σ2 ) sin (κh) 2 [ σ2 sin2 (κh)− h2 ] cos (ωσ) , (A.9b) ρ2 .= −2hσ sin2 (ωσ/2) [2 cos (κh/2)− cos (3κh/2)] sin (κh/2) [cos (κh)− cos (ωσ)] cos (ωσ) − 2ihσ sin2 (ωσ/2) sin (κh/2) sin (κh/2) [cos (κh)− cos (ωσ)] cos (ωσ) . (A.9c) As a consequence of equation (A.3) with u (0) 1 = 0, the right hand side of equation (A.9a) is secular. Hence we have to require that ∂m1u (1) 2 − σ cos (κh) h cos (ωσ) ∂n1u (1) 2 = 0, (A.10a) i∂m2u (1) 1 = B2∂ 2 ξu (1) 1 + ρ2 ∣∣u(1) 1 ∣∣2u(1) 1 . (A.10b) Equation (A.10a) implies that u(1) 2 also depends on ξ while equation (A.10b) is a nonintegrable nonlinear Schrödinger equation, as from the definition (A.9c) we can see that ρ2 is a complex coefficient. So we can conclude that equation (2.6) is not A1-integrable. Let us perform the multiple scale reduction of the Burgers equation (2.1). Equation (2.1) can be always obtained as a semicontinuous limit of equation (2.6) defining the slow times tj .= σmj , j ≥ 1. In such a way we can use in the present calculation the results presented up above. • Order ε and θ = 0: In this case the resulting equation is automatically satisfied. • Order ε and θ = 1: Taking the semi continuous limit of equation (A.1), one obtains the dispersion relation ω = −sin (κh) h . (A.11) • Order ε2 and θ = 0: Taking the semi continuous limit of equation (A.2), one obtains ∂t1u (0) 1 − 1 h ∂n1u (0) 1 = 0, (A.12) which implies that u(0) 1 depends on the variable ρ .= hn1 + t1. • Order ε2 and α = 1: Taking the semi continuous limit of equation (A.3), one obtains ∂t1u (1) 1 − cos (κh) h ∂n1u (1) 1 + 2 sin2 (κh/2)u(0) 1 u (1) 1 = 0, (A.13) which implies that u(1) 1 has the form u (1) 1 = g(1) (ξ, tj , j ≥ 2) exp { − ∫ ρ ρ0 u (0) 1 ( ρ′ ) dρ′ } , (A.14) where ξ .= hn1 + cos (κh) t1, g(1) is an arbitrary function of its arguments going to zero as ξ → ±∞ and ρ0, by a proper redefinition of g, can always be chosen to be a zero of u(0) 1 . C-Integrability Test for Discrete Equations via Multiple Scale Expansions 13 • Order ε2 and θ = 2: Taking the semi continuous limit of equation (A.5), one obtains u (2) 2 = h 1− eiκh u (1)2 1 . (A.15) • Order ε3 and θ = 0: Taking the semi continuous limit of equation (A.6), one obtains ∂t1u (0) 2 − 1 h ∂n1u (0) 2 = −∂t2u (0) 1 − 2 sin2 (κh/2) [ ∂n1 + 2hu(0) 1 ]∣∣u(1) 1 ∣∣2. (A.16) By equation (A.12), the term ∂t2u (0) 1 is a solution of the left hand side of equation (A.16), hence it is a secular term. As a consequence we have to require that ∂t1u (0) 2 − 1 h ∂n1u (0) 2 = −2 sin2 (κh/2) [ ∂n1 + 2hu(0) 1 ]∣∣u(1) 1 ∣∣2, (A.17) ∂t2u (0) 1 = 0. Solving equation (A.17) taking into account equation (A.14), we obtain u (0) 2 = f (ρ, tj , j ≥ 2) + h ∣∣u(1) 1 ∣∣2, (A.18) where f is an arbitrary function of its arguments going to zero ρ→ ±∞. • Order ε3 and θ = 1: For simplicity, from now on we require no dependence on ρ1 so that, in order to satisfy the asymptotic conditions, it necessarily follows that u (0) 1 = f = 0. (A.19) Taking the semi continuous limit of equations (A.9a), (A.9b), one obtains ∂t1u (1) 2 − cos (κh) h ∂n1u (1) 2 = −∂t2u (1) 1 − iB2∂ 2 ξu (1) 1 , B2 .= −h sin (κh) 2 . (A.20) As a consequence of equation (A.13) (with u(0) 1 = 0), the right hand side of equation (A.20) is secular. Hence we have to require that ∂t1u (1) 2 − cos (κh) h ∂n1u (1) 2 = 0, (A.21a) i∂t2u (1) 1 −B2∂ 2 ξu (1) 1 = 0. (A.21b) Equation (A.21a) implies that u(1) 2 depends also on ξ while, contrary to equation (A.10b), equation (A.21b) now is a linear Schrödinger equation, reflecting the C-integrability of equation (2.1). • Order ε3 and α = 2: Taking into account the dispersion relation (A.11), the fact that u (0) 1 = 0 and the equations (A.13), (A.15), we have u (2) 3 = h [ 2 1− eiκh u (1) 2 − h 4 sin2 (κh/2) ∂ξu (1) 1 ] u (1) 1 . (A.22) 1If ∂ρu (0) 1 6= 0, ∂ρf 6= 0, we have: u (1) 2 . = g(2) (n1, tj , j ≥ 1) exp { − ∫ ρ ρ0 u (0) 1 ( ρ′ ) dρ′ } , i∂t2g(1) = B2∂ 2 ξg(1), g(2)/g(1) = p (ξ, tj , j ≥ 2) + h [ 1 + i 2 cot ( κh 2 )] u (0) 1 + h 2 ∫ ρ ρ0 u (0)2 1 dρ′ − ∫ ρ ρ0 f ( ρ′ ) dρ′, with p arbitrary function of its arguments going to zero as ξ → ±∞. 14 C. Scimiterna and D. Levi • Order ε3 and θ= 3: Taking into account the dispersion relation (A.11) and equation (A.15), we obtain u (3) 3 = ( h 1− eiκh )2 [ u (1) 1 ]3 . (A.23) • Order ε4 and θ = 0: Taking into account equations (A.15), (A.18), (A.19), (A.21b), we get ∂t1u (0) 3 − 1 h ∂n1u (0) 3 = h∂ξ [ h 2 ∂ξ ∣∣u(1) 1 ∣∣2 − 2 sin2 (κh/2) ( u (1) 2 u (−1) 1 + u (−1) 2 u (1) 1 )] . (A.24) Solving equation (A.24), we obtain u (0) 3 = τ (ρ, tj , j ≥ 2) + h ( u (1) 2 u (−1) 1 + u (−1) 2 u (1) 1 ) − h2 4 sin2 (κh/2) ∂ξ ∣∣u(1) 1 ∣∣2, (A.25) where τ is an arbitrary function of its arguments going to zero as ρ → ±∞. As usually, if we don’t want any dependence at all from ρ but only on ξ, in order to satisfy the asymptotic conditions lim ξ→±∞ u (0) 3 = 0, we have to take τ = 0 (A.26) (unless we take the fully continuous limit). • Order ε4 and θ = 1: Taking into account the dispersion relation (A.11) and equations (A.15), (A.18), (A.19), (A.22), (A.25), (A.26), we get ∂t1u (1) 3 − cos (κh) h ∂n1u (1) 3 = −∂t3u (1) 1 − ∂t2u (1) 2 −B3∂ 3 ξu (1) 1 − iB2∂ 2 ξu (1) 2 , B3 .= −h 2 cos (κh) 6 . (A.27) As a consequence of the equations (A.13), (A.21a) and as u(0) 1 = 0, the right hand side of equation (A.27) is secular, so that ∂t1u (1) 3 − cos (κh) h ∂n1u (1) 3 = 0, (A.28a) ∂t2u (1) 2 + iB2∂ 2 ξu (1) 2 = −∂t3u (1) 1 −B3∂ 3 ξu (1) 1 . (A.28b) The first relation implies that u(1) 3 also depends on ξ while the second one, as a consequence of equation (A.21b), implies that the right hand side is secular, so that i∂t2u (1) 2 −B2∂ 2 ξu (1) 2 = 0, (A.29a) ∂t3u (1) 1 +B3∂ 3 ξu (1) 1 = 0. (A.29b) Equation (A.29a), as one can see from the definition (2.16), has a forcing term f2(2) with coefficients a = b = 0. • Order ε4 and θ = 2: Taking into account the dispersion relation (A.11) and the equa- tions (A.13), (A.15), (A.18), (A.19), (A.21), (A.22), (A.23), we get u (2) 4 = −h3 [ 1 (1− eiκh)2 u (1) 1 \|u(1) 1 \|2 + i cos (κh/2) 8 sin3 (κh/2) ∂2 ξu (1) 1 ] u (1) 1 − h2 4 sin2 (κh/2) ∂ξ ( u (1) 1 u (1) 2 ) + h 1− eiκh ( u (1)2 2 + 2u(1) 1 u (1) 3 ) . (A.30) C-Integrability Test for Discrete Equations via Multiple Scale Expansions 15 • Order ε4 and θ = 3: Taking into account the dispersion relation (A.11) and the equa- tions (A.13), (A.15), (A.19), (A.22), (A.23), we get u (3) 4 = ( h 1− eiκh u (1) 1 )2 ( 3u(1) 2 + 2heiκh 1− eiκh ∂ξu (1) 1 ) . • Order ε4 and θ = 4: Taking into account the dispersion relation (A.11) and the equa- tions (A.15), (A.23), we get u (4) 4 = ( h 1− eiκh )3 [ u (1) 1 ]4 . • Order ε5 and θ = 0: Taking into account equations (A.15), (A.18), (A.19), (A.21b), (A.22), (A.26), (A.25), (A.28), we get ∂t1u (0) 4 − 1 h ∂n1u (0) 4 = h∂ξ { − 2 sin2 (κh/2) ( u (1) 1 u (−1) 3 + u (−1) 1 u (1) 3 + \|u(1) 2 \|2 ) + [ 4 sin2 (κh/2)− 1 ] h2 2 ∣∣u(1) 1 ∣∣4 + h 2 ∂ξ ( u (1) 1 u (−1) 2 + u (−1) 1 u (1) 2 ) + ih2 4 cot (κh/2) ∂ξ ( u (−1) 1 ∂ξu (1) 1 − u (1) 1 ∂ξu (−1) 1 )} . (A.31) Solving equation (A.31), we obtain u (0) 4 = Θ(ρ, tj , j ≥ 2) + h ( u (1) 1 u (−1) 3 + u (−1) 1 u (1) 3 + \|u(1) 2 \|2 ) + h2 4 sin2 (κh/2) { ∂ξ ( u (1) 1 u (−1) 2 + u (−1) 1 u (1) 2 ) + [ 4 sin2 (κh/2)− 1 ] h ∣∣u(1) 1 ∣∣4} − ih3 cos (κh/2) 8 sin3 (κh/2) ∂ξ ( u (−1) 1 ∂ξu (1) 1 − u (1) 1 ∂ξu (−1) 1 ) , (A.32) where Θ is an arbitrary function of its arguments going to zero as ρ → ±∞. As usually, if we don’t want any dependence from ρ but only on ξ, in order to satisfy the asymptotic conditions lim ξ→±∞ u (0) 4 = 0, we have to take Θ = 0 (unless we take the fully continuous limit). • Order ε5 and θ = 1: Taking into account the dispersion relation (A.11) and equations u (0) 1 = f = τ = Θ = 0, together with the equations (A.15), (A.18), (A.22), (A.23), (A.25), (A.30), (A.32), we get ∂t1u (1) 4 − cos (κh) h ∂n1u (1) 4 = −∂t4u (1) 1 − ∂t3u (1) 2 − ∂t2u (1) 3 − iB2∂ 2 ξu (1) 3 −B3∂ 3 ξu (1) 2 + iB4∂ 4 ξu (1) 1 + ζ [ u (1) 1 ∣∣∂n1u (1) 1 ∣∣2 + u (−1) 1 ( ∂n1u (1) 1 )2 + u (1)2 1 ∂2 n1 u (−1) 1 ] , (A.33) B4 .= h3 sin (κh) 24 , ζ .= h [1 + cos (κh) + 3i sin (κh)] eiκh − 1 . As a consequence of the equations (A.13), (A.21a), (A.28a) and of u(0) 1 = 0, the right hand side of equation (A.33) is secular, so that ∂t1u (1) 4 − cos (κh) h ∂n1u (1) 4 = 0, 16 C. Scimiterna and D. Levi ∂t2u (1) 3 + iB2∂ 2 ξu (1) 3 = −∂t4u (1) 1 − ∂t3u (1) 2 −B3∂ 3 ξu (1) 2 + iB4∂ 4 ξu (1) 1 + ζ [ u (1) 1 ∣∣∂n1u (1) 1 ∣∣2 + u (−1) 1 ( ∂n1u (1) 1 )2 + u (1)2 1 ∂2 n1 u (−1) 1 ] . (A.34) The first relation tells us that u(1) 4 depends on ξ too while in the second one, as a con- sequence of equations (A.21b), (A.29a), the first our terms in the right hand side of equation (A.34) are secular, so that ∂t2u (1) 3 + iB2∂ 2 ξu (1) 3 = ζ [ u (1) 1 ∣∣∂n1u (1) 1 ∣∣2 + u (−1) 1 ( ∂n1u (1) 1 )2 + u (1)2 1 ∂2 n1 u (−1) 1 ] , (A.35a) ∂t3u (1) 2 +B3∂ 3 ξu (1) 2 = −∂t4u (1) 1 + iB4∂ 4 ξu (1) 1 . (A.35b) As a consequence of equation (A.29b), the right hand side of equation (A.35b) is secular so that ∂t3u (1) 2 +B3∂ 3 ξu (1) 2 = 0, ∂t4u (1) 1 − iB4∂ 4 ξu (1) 1 = 0. Equation (A.35a), as one can see from the definition (2.17) and taking into account that a = b = 0, has a forcing term f2(3) that respects all the A3 C-integrability con- ditions (2.18). 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C-Integrability Test for Discrete Equations via Multiple Scale Expansions

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